255

Critical Studies/Book Reviews

David Hilbert. ’s lectures on the foundations of , 1891–1902. Michael Hallett and Ulrich Majer, eds. David Hilbert’s Foun- Downloaded from dational Lectures; 1. Berlin: Springer-Verlag, 2004. ISBN 3-540-64373-7. Pp. xxviii + 661.† Reviewed by Victor Pambuccian∗ http://philmat.oxfordjournals.org/ 1. Introduction This volume, the result of monumental editorial work, contains the German text of various lectures on the , as well as the first edition of Hilbert’s Grundlagen der Geometrie, a work we shall refer to as the GdG. The material surrounding the lectures was selected from the Hilbert papers stored in two Gottingen¨ libraries, based on criteria of relevance to the genesis and the trajectory of Hilbert’s ideas regarding

the foundations of geometry, the rationale, and the meaning of the foun- at Arizona State University Libraries on December 3, 2013 dational enterprise. By its very nature, this material contains additions, crossed-out words or paragraphs, comments in the margin, and all of it is painstakingly documented in notes and appendices. Each piece is pre- ceded by informative introductions in English that highlight the main fea- tures worthy of the reader’s attention, and place the lectures in historical context, sometimes providing references to subsequent developments. It is thus preferable (but by no means necessary, as the introductions alone, by singling out the main points of interest, amply reward the reader of English only) that the interested reader be proficient in both German and English (French would be helpful as well, as there are a few French quotations), as there are no translations. Chapter 1 consists of manuscript notes in Hilbert’s own hand for a course on held in Konigsberg¨ in the summer semester of 1891; Chapter 2 is his own notes for a course on the foundations of geometry in Konigsberg,¨ planned for the summer of 1893, but held during the summer semester of the following year; Chapter 3 contains Hilbert’s notes for two Ferienkurse for high-school teachers which took place in Gottingen¨ in 1896 and 1898. Perhaps the most interesting of all, is Chapter 4. It consists of two versions of notes for a course on the

† I thank John Baldwin, Michael Beeson, Robin Hartshorne, Marvin Greenberg, and Horst and Rolf Struve for their criticism, suggestions, and corrections of an earlier version. ∗ School of Mathematical and Natural Sciences, Arizona State University — West Cam- pus, Phoenix, Arizona 85069-7100, U.S.A. [email protected].

Philosophia Mathematica (III) Vol. 21 No. 2 C The Author [2013]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected] 256 PHILOSOPHIA MATHEMATICA foundations of , delivered during the winter semester 1898/1899 in Gottingen.¨ These notes form the basis of the GdG of 1899, and contain a wealth of information of a strictly mathematical nature, which never made it into any edition of the GdG, and a wealth of state- ments relevant to both Hilbert’s motivations for proceeding the way he does and his philosophical views on the nature of the axiomatic enterprise. Downloaded from Unlike the Hilbert of GdG, with his extreme concision, the Hilbert we encounter in these lectures is talkative and opens a window to his mathe- matical mind’s house, allowing us glimpses of his aims and thoughts, and their gradual change with the passage of time. Chapter 5 contains the text of the first edition of the GdG, — a book that has become, like many first http://philmat.oxfordjournals.org/ editions, a rarity1 — referred to as the Festschrift, as it was written on the occasion of the unveiling of the Gauss-Weber monument. The introduction to Chapter 5 refers succinctly to the various editions that followed, with- out going into details regarding the changes. The reader who would like to have a synopsis of all these changes, all the way up to the eighth edition, should consult [Hilbert, 1971]. Chapter 6, the last one, contains a version of Hilbert’s lectures on the foundations of geometry delivered during the summer semester of 1902. This offers all interested researchers material that has been first interpreted in [Toepell, 1986], but had been available at Arizona State University Libraries on December 3, 2013 only through selected quotations. We will focus in this review on the following topics: (i) Hilbert’s view on the process of axiomatization and on the nature of geometry, as can be read from the material in the lectures; (ii) the specific questions raised in the lectures as providing a road map for the advances in the foundations of geometry obtained during the twentieth century and beyond; (iii) interest- ing material that can be found in the lectures, but not in GdG; and (iv) the legacy of the GdG.

2. The Process of Axiomatizaton and the Nature of Geometry Ad (i). First of all, one finds that Hilbert is very far in practice from formalism in the wide sense (as emphasized by Corry [2000; 2002; 2006] repeatedly) in which becomes a game with strict rules that manipulates meaningless formulas. His formalism is only a tool by which he tried to prove the consistency of arithmetic. In the practice of the axiomatization of geometry he did what his predecessors, Pasch, Peano, Pieri, had done as well, namely imagine an axiom system that does not require an understanding of either the entities or the predicates of the lan- guage in which it is expressed, that would go beyond having grasped their

1 It will likely become even more of a rarity, as university libraries in the United States have been discarding it, given that it was superseded by newer editions. One such discarded copy of the Festschrift was saved in the twenty-first century by Alexander Soifer for US$1. CRITICAL STUDIES/BOOK REVIEWS 257 characteristics as stipulated by the axioms of the axiom system. In fact, one is surprised to find that, although his empiricism is not as strict as that of Pasch — whose empiricism has recently been the subject of intense scrutiny [Gandon, 2005; Schlimm, 2010] — he shares with Pasch the view that geometry corresponds to aspects of our experience. On page 171 he notes, for example, that the and the Archimedean Axiom Downloaded from do not have the empirical2 character of the other axioms that allow con- struction by a finite number of experiments. On page 302 we read that ‘geometry is the the most accomplished natural science’,3 on page 303 that the task of the foundations of geometry is that of ‘a logical analysis of our intuition’s capabilities’,4 on page 391 that ‘the geometric theorems http://philmat.oxfordjournals.org/ are never true in nature in full exactness, since the axioms are never ful- filled by objects’, but that this should not be taken to be a flaw, for it is a characteristic of theories, for ‘a theory, that would coincide in all details with reality, would be no more than an exact description of the objects’. Another remarkable feature of Hilbert’s approach to the axiomatization of Euclidean geometry is the high regard which he has for Euclid, whom he is proud to correct or outwit sometimes. The entire enterprise, of turning geometry back into the deductive art it once was in ancient Greece, away from the ars calculandi that the analytic geometry ‘premisses’ of another at Arizona State University Libraries on December 3, 2013 age had turned geometry into,5 appears to have taken a page from Johann Joachim Winckelmann, whose credo was ‘Der einzige Weg f¨ur uns, groß, ja, wenn es moglich¨ ist, unnachahmlich zu werden, ist die Nachahmung der

2 Hilbert’s own emphasis. 3 ‘Geometrie ist die vollkommenste Naturwissenschaft.’ 4 ‘eine logische Analyse unseres Anschauungsvermogens’.¨ On the first page of the GdG itself, we find that the task of the book amounts to ‘the logical analysis of our spatial intuition.’ (‘die logische Analyse unserer raumlichen¨ Anschauung’ (p. 436)) 5 ‘Mit diesen Pramissen¨ ist dann sofort aus der Geometrie eine Rechenkunst geworden’ (p. 222). By creatively restoring elementary geometry to its former state, to an undertak- ing confined to its own language, without algebraic admixtures, Hilbert finds a solution to the critique expressed often by Newton of the Cartesian approach (for an in-depth look at Newton’s pronouncements regarding geometry and algebra see [Guicciardini, 2009]), such as in: ‘Anyone who examines the constructions of problems by the straight line and circle devised by the first geometers will readily perceive that geometry was contrived as a means of escaping the tediousness of calculation by the ready drawing of lines. Consequently these two sciences ought not to be confused. The Ancients so assiduously distinguished them one from the other that they never introduced arithmetical terms into geometry; while recent people, by confusing both, have lost the simplicity in which all elegance of geometry consists’ [Newton, 1673/83, p. 428]. Hilbert expresses a similar view in the 1898–99 lec- tures: ‘However, if science is not to fall prey to a sterile formalism,thenitmust reflect in a later stage of its development on itself, and should at least analyse the foundations that led to the introduction of number.’ (‘Aber, wenn die Wissenschaft nicht einem unfruchtbaren Formalismus anheimfallen soll, so wird sie auf einem spateren¨ Stadium der Entwicklung sich wieder auf sich selbst besinnen mussen¨ und mindestens die Grundlagen pr¨ufen,auf denen sie zur Einf¨uhrungder Zahl gekommen ist.’ (p. 222)) 258 PHILOSOPHIA MATHEMATICA

Alten.’6 Referring to the relationship between projective and Euclidean geometry, Hilbert finds that ‘in projective geometry one takes recourse from the very beginning to intuition’, whereas in the axiomatization of Euclidean geometry it is precisely ‘intuition that we want to analyze, to build it afterwards anew out of its constituent parts’.7 Perhaps surprising in view of Hilbert’s later positions, but understandable in a time in which Downloaded from the notion of first-order was still more than two decades away, is the fact that, in 1898–1899, the logical background against which an axiom system is set up, consists not just of logic, but rather of ‘the laws of pure logic as well as the entire arithmetic’,8 with a reference to Dedekind’s Was sind und was sollen Zahlen? regarding the relationship between logic http://philmat.oxfordjournals.org/ and arithmetic, as if in agreement with the logicist stance on the issue.9 Something resembling arithmetic, however, is used only in the formula- tion of the Archimedean Axiom, for one of the aims of the GdG is to intro- duce number into geometry in a purely geometrical manner, for ‘in any exact science, a most highly prized aim is the introduction of numbers’.10 Hilbert’s obsession with the ‘introduction of number’ (‘Einf¨uhrung der Zahl’), which he finds (on p. 282) to be one of those ‘true, genuine won- ders’ Lessing’s Nathan the Wise referred to as miraculously becoming rou- tine, leads us to our next topic, (ii). at Arizona State University Libraries on December 3, 2013

3. A Road Map for the Advances in the Foudations of Geometry We will look at the main questions Hilbert asked in his lecture notes, and the way in which they dominated much of the research in the axiomatic foundation of geometry. Since these questions revolve around two configuration theorems, Desargues and Pappus, we first state them. The projective form of the Desargues theorem (to be denoted by Des) states that ‘If ABC and A BC are two triangles, such that the lines AA, BB, CC meet in a O, then the intersection points X, Y, Z of the line pairs AB and A B, BC and BC, CAand C A are collinear.’ (The affine form (to be denoted by a Des) states that if two of the line pairs above are parallel, then so is the third pair.) The projective form of Pappus

6 ‘The only way in which we could become great, and even, if at all possible, inimitable, is the imitation of the ancients.’ 7 ‘in der projectiven Geometrie appelliert man von vornherein an die Anschauung, wogegen wir ja die Anschauung analysieren wollen, um sie dann sozusagen aus ihren Bestandteilen wieder aufzubauen.’ (p. 303) 8 ‘Als gegeben betrachten wir die Gesetze der reinen Logik und speciell die ganze Arith- metik.’ (p. 303) 9 Ferreiros´ [2009] believes that Hilbert actually was a logicist at the time. For a compar- ison of Dedekind’s and Hilbert’s early approach to foundational matters see [Klev, 2011]. 10 ‘Nun ist in der That in jeder exakten Wissenschaft die Einf¨uhrung der Zahl ein vornehmstes Ziel.’ (p. 222) CRITICAL STUDIES/BOOK REVIEWS 259

(to be denoted by Papp) states that ‘If A, B, C and A, B, C are two sets of three collinear points, lying on different lines, then the intersections points X, Y, Z of the line pairs AB and A B, AC and AC, BC and BC are collinear’. (The affine version (to be denoted by a Papp) states that if two of the line pairs above are parallel, then so is the third pair.) Notice that the projective forms are universal statements in terms of points, lines, Downloaded from and incidence alone, as the existence of the points of intersection of the respective lines is part of the hypothesis of these configuration theorems. Hilbert’s recurring questions are: (1) Is a Des provable with the help of the congruence axioms alone? http://philmat.oxfordjournals.org/ (p. 172) (2) Does a Papp (referred to as Pascal’s theorem throughout) follow from the congruence axioms together with a Des? (p. 174) (3) Does a Des follow from the congruence axioms and a Papp? (pp. 174, 175) (4) Prove a Papp (and a Des) in the plane based only on axioms in the groups I, II, and III (i.e., on the basis of the axioms for , without using the Parallel Postulate) (pp. 284, 392); in another form: a Papp ‘arises by the elimination of the congruence at Arizona State University Libraries on December 3, 2013 axioms, indeed [a Papp] is the sufficient condition that ensures that a definition of congruence is possible’. (p. 261)11 (5) a Des must hold if a plane is to be part of space.12 Is it also a suffi- cient condition for this to happen? (p. 318) (6) ‘The Desargues theorem is the result of the elimination of the spatial axioms from I and II’ (p. 318).13 Also in connection with a Des, Hilbert expressed a much broader14 con- cern that, we are told on the last page of the GdG, is the ‘subjective form’ of the main concern of the axiomatic foundation of geometry,15 namely

11 Here Hilbert refers to both (4) and (6) in the same sentence. It reads: ‘So wie der Desargues gewissermassen die Elimination der raumlichen¨ Axiome ist, so entsteht der Pascal durch die Elimination der Congruenzaxiome und zwar ist der Pascal auch die hinre- ichende Bedingung daf¨ur, dass eine Congruenzdefinition moglich¨ ist.’ (p. 261) 12 Notice that Hilbert had proved the necessity (and sufficiency) of a Des for a plane to be part of three-dimensional space only in the affine case. That any plane in an ordered space (i.e., in a model of Hilbert’s incidence and order axioms) has to satisfy Desargues wasfirstshownin[Pasch, 1882, pp.46–55] (see also, [Abellanas, 1946]) 13 Since there are no proper spatial axioms in the group II of order axioms, Hilbert must have meant here just group I of incidence axioms. 14 See also [Arana, 2008; Arana and Detlefsen, 2011, pp. 6–7]. 15 ‘Der Grundsatz, demzufolge man ¨uberalldie Principien der Moglichkeit¨ der Beweise erlautern¨ soll, hangt¨ auch aufs Engste mit der Forderung der “Reinheit” der Beweismetho- den zusammen, die von mehreren Mathematikern der neueren Zeit mit Nachdruck erhoben worden ist. Diese Forderung ist im Grunde nichts Anderes als eine subjektive Fassung 260 PHILOSOPHIA MATHEMATICA

(7) the concern with the purity of the method (‘Reinheit der Methode’), stemming from the fact that a Des, although a statement belonging to plane geometry, is usually proved by recourse to the geometry of space, in which the plane is supposed to be embedded. Here Hilbert criticizes the means of proof (although he shows that a Des Downloaded from cannot be proved from the plane incidence axioms — not even if the con- gruence axioms, with the ‘Side-Angle-Side’ congruence axiom omitted, and the Euclidean Parallel Postulate are allowed in the proof) and states that ‘In modern mathematics one exercises this sort of critique very often, whereby we notice the tendency to preserve the purity of the method, i.e., http://philmat.oxfordjournals.org/ to use in the proof of a theorem, if possible, only the auxiliary means that the content of the theorem imposes upon us.’16 (8) Is the Archimedean axiom needed to prove the Legendre theorems (regarding two facts, namely that if the sum of the angles of one triangle is π,thenitisπ for all triangles, and that the sum of the angles of every triangle does not exceed π)? (p. 392) All of these questions were prompted by the interest in ‘introducing num- ber’ into geometry, which Hilbert finds not only of interest for purity of the at Arizona State University Libraries on December 3, 2013 method’s sake, but also as a means of transferring the consistency prob- lem of Euclidean geometry to that of a ‘number system’, a certain field — or a certain class of fields — in today’s language,17 for the Desargues theorem (in both forms) is a configuration theorem (i.e., a universal state- ment regarding points, lines, and incidence) that ensures that the plane can be coordinatized by means of skew fields (being essential in proving the associativity of multiplication), and the Pappus theorem (in both forms) is a configuration theorem ensuring the commutativity of multiplication. These problems have motivated a great part of the axiomatic founda- tions in the twentieth century and beyond, and have produced some of des hier befolgten Grundsatzes. In der That sucht die bevorstehende Untersuchung allge- mein dar¨uberAufschluss zu geben, welche Axiome, Voraussetzungen oder Hilfsmittel zum Beweise einer elementar-geometrischen Wahrheit notig¨ sind’ (p. 523; last page (page 90) of the GdG). 16 In der modernen Mathematik wird solche Kritik sehr haufig¨ ge¨ubt, wobei das Bestreben ist, die Reinheit der Methode zu wahren, d.h. beim Beweise eines Satzes wo moglich¨ nur solche H¨ulfsmittelzu benutzen, die durch den Inhalt des Satzes nahe gelegt sind. (pp. 315–316) 17 The problem regarding the consistency of solid Euclidean geometry is addressed both in the 1898–99 lectures: ‘Diese [die analytische Geometrie] ist moglich,¨ weil die Eigen- schaften der reellen Zahlen sich nicht einander widersprechen, sondern alle miteinander vertraglich¨ sind.’ (p. 282), and in GdG — in Kap. II, §9 (pp. 454–455) — where the con- sistency of Euclidean geometry is shown by pointing to the absence of contradiction inside the Pythagorean closure of the field of rational numbers, or put differently, by assuming that the theory of Archimedean ordered Pythagorean fields is consistent. CRITICAL STUDIES/BOOK REVIEWS 261 the deepest and most astonishing results pure geometry has ever seen.18 First, let us look at the legacy of questions (1)–(4). That Papp implies Des in projective planes (i.e., if one assumes that every line is incident with at least three points, and that any two lines intersect, in addition to Hilbert’s plane incidence axioms) was proved by Hessenberg in [1905a] (a gap in the proof was closed in [Cronheim, 1953]). Whether an ordered Downloaded from plane satisfying Papp must satisfy Des as well (i.e., whether the plane inci- dence and order axioms together with Papp imply Des) is still not known. The first contribution regarding the effect of the congruence axioms (in the absence of the Parallel Postulate or the hyperbolic version thereof) on Papp and Des, is in the ground-breaking paper by Hjelmslev [1907]. http://philmat.oxfordjournals.org/ The central realization of that work was that line-reflections have cer- tain properties that are independent of any assumption regarding par- allels and thus absolute. Line-reflections, and in particular the central three-reflection theorem, stating that the composition of three reflections in lines that have a common perpendicular or a common point must be a line-reflection, had appeared earlier in [Wiener, 1893; Schur, 1899; Hilbert, 1903a; Hessenberg, 1905b], but in these works line-reflections were not treated independently of the particular geometry in which they were defined (Euclidean, hyperbolic, or elliptic), as they were by Hjelm- at Arizona State University Libraries on December 3, 2013 slev, who carried on this research in [Hjelmslev, 1929]. Many more — whose contributions are chronicled in [Karzel and Kroll, 1988] — have helped understanding geometry in terms of line-reflections as primitive notions. They helped remove assumptions regarding both the order of the plane and the free mobility of the plane (i.e., the possibility of transport- ing segments on any given line). What is left after the removal work was done consists of the three-reflections theorem, together with very basic axioms stating that there are at least two points, that there is exactly one line incident with two distinct points, that perpendicular lines intersect, and that through every point there is a perpendicular to any line, which is unique if the point and the line are incident. The final touch in carving this austere axiom system came from Friedrich Bachmann [1951], who showed that two axioms proposed by Hilbert’s student Arnold Schmidt [1943] are superfluous. The axiom system can also be conceived as one expressed in terms of orthogonality alone (see [Pambuccian, 2007a]), and, in the non-elliptic case, in terms of incidence and segment congru- ence (or just in terms of the latter alone), as required by the original statement of (4), as shown in [Sorensen¨ , 1984].19 The group-theoretical and the traditional geometric axiomatizations are logically equivalent,

18 These problems are still open for many Cayley-Klein . (see [Struve and Struve, 1985; 2004; 2010].) 19 Something that Hilbert apparently considered impossible in 1898–99: ‘Der umgekehrte Weg, die Kongruenzaxiome und -satze¨ mit H¨ulfe des Bewegungsbegriffs zu 262 PHILOSOPHIA MATHEMATICA as spelled out in [Pambuccian, 2005b]. What is remarkable about this austere axiom system for structures called metric planes is that it is strong enough to prove both Papp and Des, metric planes being embed- dable in projective planes coordinatized by fields of characteristic =2 and endowed with an orthogonality relation extending that of the metric plane [Bachmann, 1973]. In fact, an even weaker axiom system, in which Downloaded from the three-reflections theorem is weakened, implies both Des and Papp. That it implies Des was shown by Sperner [1954], who set out to answer precisely the question asked by Hilbert in (1), going beyond the origi- nal question by asking for minimal ‘congruence’ assumptions, expressed in terms of line-reflections.20 It turns out that even Sperner’s structures, http://philmat.oxfordjournals.org/ which were thoroughly studied in [Lingenberg, 1959; 1960/1961; 1965], satisfy Papp (see [Karzel and Kroll, 1988, pp. 181–182] and [Lingenberg, 1979]), and they can be embedded in projective planes satisfying Papp. A far-reaching generalization of Sperner’s structures was proposed in [Schroder¨ , 1984], and studied in depth in [Saad, 1988]. No axiom system for some absolute geometry is known that would satisfy only Des, but not Papp. There are several axiom systems [Sch¨utte, 1955a; 1955b; Naumann, 1956; Naumann and Reidemeister, 1957; Quaisser, 1975; Szmielew, 1983;

Kusak, 1987] that provide in the Euclidean case a precise answer to (1) at Arizona State University Libraries on December 3, 2013 and (2), i.e., provide orthogonality or congruence axioms that are strong enough to imply a Des but not a Papp. In the same Desargues and Pappus area of concern, we find the following statements of Hilbert that ‘Every configuration theorem can be proved by means of Pappus and Desargues using only the incidence axioms.’21 This statement was included in the GdG and preserved in all future editions. It states that every configura- tion theorem true in ordered affine planes in which a Des and a Papp hold (later the requirement that a Des holds is dropped, with a reference to [Hessenberg, 1905b]) ‘will always turn out to be a combination of the Desargues and Pappus theorems’ (GdG, Kap. VI, §35, p. 511), and in later editions ‘a combination of finitely many Pappus configurations’. If by ‘configuration theorem’ we are to understand a universal sentence in terms of points, lines, and point-line incidence, then it is not true that every configuration theorem true in ordered Pappian affine planes can be beweisen ist falsch, da sich die Bewegung ohne den Kongruenzbegriff gar nicht definieren lasst.’¨ (p. 335) 20 In fact, given that its models can be embedded in projective planes over commutative fields without the requirement that the characteristic be =2, the axiom system in [Sperner, 1954] answers a minimalist question very close to (1)–(4), namely that for ‘congruence’ axioms that would, together with the trivial incidence axioms, be just strong enough to prove Papp and every projective configuration theorem that can be proved from Papp, but no other configuration theorem. 21 ‘Aus Pascal und Desargues kann allein durch die Axiome der Verkn¨upfungjeder Schnittpunktsatz [...]bewiesenwerden.’(p.178) CRITICAL STUDIES/BOOK REVIEWS 263 derived from the incidence axioms for affine planes and a Papp,forthe obvious reason that Fano’s axiom (‘The three diagonal points of a com- plete quadrilateral are never collinear’) must hold in ordered affine planes, and Fano’s axiom is not true in all Pappian affine planes. This was certainly clear to Hilbert. Whatever he might have had in mind, his having empha- sized the universal incidence theory of a theory expressed in a language Downloaded from with incidence and order, raised a general question with far-reaching con- sequences. Given a theory TBI in terms of betweenness B and incidence I , find its universal incidence theory (TI )∀. With TBI being the theory of ordered affine planes satisfying a Papp, the question was answered in

[Artin and Schreier, 1926]. The answer is: one needs to add to the trivial http://philmat.oxfordjournals.org/ a incidence axioms and to Papp an axiom schema stating, in (TI )∀, that the sum of non-zero squares is never equal to zero. In the case of ordered affine planes satisfying a Des the answer was provided independently in [Pickert, 1951]and[Szele, 1952], and for the plain theory of ordered affine planes in [Kalhoff, 1988]. The results of Sperner [1938]andJoussen [1966] deter- mine the theories (TI )∀ even if TBI is the theory of ordered planes satisfy- ing Des or just that of plain ordered planes (axiomatized by Hilbert’s plane incidence and order axioms).

As far as (5) and (6) are concerned, Hilbert provides with Theorem at Arizona State University Libraries on December 3, 2013 35 of GdG (Kap. V, §30, p. 505) a rather modest answer to the ques- tions. For although he shows that Desarguesian ordered affine planes can be embedded in ordered affine spaces, and thus that a Des is indeed a sufficient condition for a plane to be a part of space, he does this in an algebraic manner. After coordinatizing the affine plane by a skew field, he adds a third coordinate to each point (x, y) of the plane, which becomes (x, y, 0) in the affine space in which the plane is embedded. A proof more geometrico of Theorem 35 was provided in [Schor, 1904]. Purely geometric embeddings of any Desarguesian projective plane into a 3-dimensional projective space (without any order relation) were pro- vided in [Levi, 1939; Fritsch, 1974; Herzer, 1975; Baldwin, 2013], the proof by Howard and Baldwin in [Baldwin, 2013, §4] being entirely in the spirit of classical projective geometry. The first step toward the solu- tion of (5) in the case of ordered planes, was taken by Owens [1910], who showed that an ordered plane, in which a strong form of Desargues axiom (the point O does not need to be a proper point) and of its con- verse hold, can be embedded in a projective ordered plane satisfying Des. Unaware of Owen’s work, Ruth Moufang [1931] showed that an ordered plane that satisfies the projective minor Desargues axiom (whenever all points involved belong to the ordered plane) can be embedded in a projec- tive ordered plane, which in turn satisfies the projective minor Desargues axiom. Sperner [1938] completely solved Hilbert’s problems (5) and (6) by showing that Desarguesian ordered planes can be embedded in pro- jective Desarguesian ordered planes (it follows from results proved later, 264 PHILOSOPHIA MATHEMATICA in [Skornyakov, 1949]and[Bruck and Kleinfeld, 1951], that Moufang’s 1931 result implies Sperner’s 1938 result), and thus that they are indeed part of ordered three-dimensional spaces as defined in GdG. Problem (7) has opened up a large field of investigations, by asking for pure proofs for a specific result, which amounts to proofs proceeding from minimalist assumptions, both in the sense of the language employed and in Downloaded from the sense of a minimalist content of the axioms. This does not boil down to searching for a proof in absolute geometry for a theorem known to be true in Euclidean geometry, but rather aims at finding the right assumptions that are needed to prove a theorem. In Hilbert’s own words http://philmat.oxfordjournals.org/ By the axiomatic analysis of a mathematical truth I understand an investigation, which does not aim to discover new or more general theorems relative to that truth, but rather aims to clarify the position of that theorem inside the system of known truths and their mutual logical connections in such a way that one can indicate exactly which conditions are necessary and sufficient for justifying that truth.22

Perhaps the first such in-depth investigation was Hilbert’s own [1903b]

(to become Appendix II to GdG in later editions), followed by the search at Arizona State University Libraries on December 3, 2013 for the minimalist ‘congruence’ (in fact, orthogonality) axioms required to prove Pappus, resulting in the flurry of papers mentioned earlier regarding (4), by the in-depth analysis of the axioms required to prove the Mobius-Pompeiu¨ theorem from [Barbilian, 1936] (see also [Pambuccian, 2009a]), by Coxeter’s considerations on the purity of the proof of the Sylvester-Gallai theorem (see also [Pambuccian, 2009b]), by the mar- velous result from [Bachmann, 1967] regarding the minimal assumptions required to prove the generalized concurrency of the altitudes of a triangle, by the analysis of the axioms needed to prove Euclid’s Proposition I.1 in [Pambuccian, 1998a] (see also [Hartshorne, 2000, p. 373]), as well as in [Schroder¨ , 1985; Fritsch, 1995; Pambuccian, 2003; 2005a; 2006; 2007b; 2010; Hociota˘ and Pambuccian, 2011; Pambuccian and Struve, 2009; Pambuccian, Struve, and Struve, forthcoming; Pambuccian, 2012], in which minimal assumptions in both primitive notions and axioms were found for specific geometric statements.23 A concerted effort to find minimal assumptions, motivated by the ‘purity of the method’ concern,

22 ‘Unter der axiomatischen Erforschung einer mathematischen Wahrheit verstehe ich eine Untersuchung, welche nicht dahin zielt, im Zusammenhange mit jener Wahrheit neue oder allgemeinere Satze¨ zu entdecken, sondern die vielmehr die Stellung jenes Satzes innerhalb des Systems der bekannten Wahrheiten und ihren logischen Zusammenhang in der Weise klarzulegen sucht, daß sich sicher angeben lat,¨ welche Voraussetzungen zur Begr¨undung jener Wahrheit notwendig und hinreichend sind.’ [Hilbert, 1903b] 23 The assumptions in [Bachmann, 1967; Pambuccian, 2003; 2006; 2009b; 2010; Hociota˘ and Pambuccian, 2011, Pambuccian, 2012] are too weak to allow for an alge- braization, providing one more reason why geometry cannot be reduced to algebra. CRITICAL STUDIES/BOOK REVIEWS 265 can be found in the papers [Andreka,´ Madarasz,´ and Nemeti´ , 2006; Andreka,´ Madarasz,´ Nemeti,´ and Szekely´ , 2008; Madarasz,´ Nemeti,´ and Szekely´ 2006; Szekely´ , 2010], focusing on very austere axioms needed to prove certain theorems in relativity. The original concern in (7), regarding Desargues’s plane nature and solid proof, is still [Arana and Mancosu, 2012; Baldwin, 2013] motivating Downloaded from reflections regarding various ways in which the purity of the method can be understood, and the relationship between plane and solid geometry. As for question (8), it was answered by [1900] a short time after it was raised, yet the crowning achievement, which would allow a wide class of like-minded questions to be answered with a modest amount http://philmat.oxfordjournals.org/ of algebraic skill, came with the algebraic characterization of all models of Hilbert’s plane axioms of incidence, order, and congruence by Wolfgang Pejas [1961], a characterization made possible by the most sophisticated answers given to (2), in particular by Bachmann [1973].

4. Previously Unpublished Material in Hilbert’s Lectures Regarding (iii), it is surprising how much more is to be found in the lec- tures than in GdG or in any other published material. Hilbert provides at Arizona State University Libraries on December 3, 2013 a wealth of independence models for all sorts of axioms, starting with the incidence axioms (pp. 306–307) and each order axiom, including an independence model for the Pasch axiom (pp. 232, 311–312), which is shown to be independent of all the incidence axioms and all the linear- order axioms. The most intriguing among the questions on independence is the question regarding the independence of the symmetry axiom for the metric, i.e., of the statement that segment AB is congruent to segment BA. Its intrinsic difficulty is responsible for the greatest confusion that can be found in the entire book. An independence model from all other axioms, excluding the Side-Angle-Side triangle-congruence axiom, is provided on page 286 (and the independence is mentioned again on p. 399), although Hilbert tries unsuccessfully to prove it twice, once on p. 10924 and then again on pp. 320–321, where he gives up the attempt (after claiming that it can be proved by ‘the introduction of a minor change’25), to re-introduce AB ≡ BAas an axiom on p. 322 (it had been taken as an axiom on p. 319). Pages 218–219 of the editors’ notes are devoted to this axiom, and on p. 218 the editors state that AB ≡ BA ‘is a simple consequence of the Triangle Congruence Axiom’ (by which the Side-Angle-Side congruence axiom is meant). It is not clear what is meant by this comment, as no proof of AB ≡ BA from a traditional axiom system is known. As far as

24 We find the following mesmerizing insertion between the proposition and its proof: ‘Dieser Satz lasst¨ sich besonders gut durch das Experiment pr¨ufen, ist jedoch oben der Einfachheit wegen unter die Axiome genommen.’ (p. 109) 25 ‘nach Anbringung einer kleinen Modifikation’ (p. 321). 266 PHILOSOPHIA MATHEMATICA

I know, the independence of AB ≡ BA is open in both Tarski’s (as stated in [Gupta, 1965, p. 40]) and in Hilbert’s axiom system. AB ≡ BA has recently been shown dependent in a variant of Tarski’s axiom system in which in the conclusion to the five-segment axiom (the version of the Side-Angle-Side triangle congruence theorem in the absence of the angle- congruence notion) the endpoints of a segment have been switched (see Downloaded from [Makarios, 2013]). It is also worth noticing that in the lectures of 1898–99 we find (much as in [Greenberg, 2008, pp. 110–111]) a plane separation axiom instead of the Pasch axiom. The latter is proved from the plane separation axiom, where we also find a proof of the space separation theorem (p. 233). http://philmat.oxfordjournals.org/ An even more interesting theorem that appears in all lecture notes from 1896 on (pp. 174, 257–258, 335–337), is the Three-Chord Theorem,which states that, if three circles pairwise intersect in two points, then the three lines joining those two points (to be referred to in the sequel as ‘chords’) are concurrent. Several proofs are presented, some algebraic, depending on Pappus, others by considering the three circles as belonging to three spheres. Hilbert’s proofs are far from enlightening; he mentions a depen- dence of some proofs on the circle-continuity principle (i.e., the fact that the co-ordinate field is Euclidean), and it is in general not clear what role at Arizona State University Libraries on December 3, 2013 he had in mind for this theorem. It follows from [Hartshorne, 2003], as pointed out by Marvin Greenberg [2010], that the Three–Chord Theorem, in the form stating that if two of the chords intersect, then the three chords must be concurrent, holds in plane absolute geometry (i.e., in planes satis- fying Hilbert’s incidence, betweenness, and congruence axioms).26 At any rate, the Three-Chord Theorem ought to be true, with concurrence replaced by the requirement that the three chords lie in a pencil, given that it is a uni- versal statement that can be stated in terms of incidence and congruence or in terms of orthogonality alone27 in Bachmann’s metric planes. If we con-

26 From [Hartshorne, 2003] it follows only that the Three-Chord Theorem holds in abso- lute planes that satisfy the line-circle continuity axiom (that states that a line that is incident with a point in the interior of a circle must intersect that circle). Universal statements being hereditary, the Three-Chord Theorem must hold in any submodel as well, and all abso- lute planes are submodels of absolute planes satisfying the line-circle continuity axiom, as every Pythagorean field has a Euclidean hull. 27 It can be stated as    = ∧ 1 = 2 ∧ n Oi O j Pij Pij Oi Pjk i< j 1≤i< j≤3 1≤i≤3,1≤ j

Chord Theorem holds, a very far cry from the requirement that the field be Downloaded from ordered and Euclidean, i.e., that every positive element be a square.)

5. The Legacy of the GdG Why is GdG important, why is the first edition deserving a reprint? Is it http://philmat.oxfordjournals.org/ just because it has been repeated often and at regular intervals that it is a classic? GdG has also had its fair share of critics, beginning with Poincare´ and ending with Freudenthal [1957].29 From a social point of view, its major importance has undoubtedly been that of turning work in the foun- dations of geometry into a socially respectable occupation, given on the one hand that no less than Hilbert himself showed interest in it, and on the other that a school pursuing work in the foundations of geometry sprung up around Hilbert.30 We will not try to explain the apparent historical suc- cess of GdG, with its fourteen German editions, given that success with an at Arizona State University Libraries on December 3, 2013 audience relies on a variety of social factors and is not always correlated with intrinsic value. What we would like to point out here is its importance sub specie æternitatis. Several historical studies have shown that much of the perceived pioneering quality of GdG is based on an ignorance of the predecessors of GdG. Freudenthal [1957] points out that the ‘introduc- tion of number’ was the work of Karl G.C. von Staudt (in his Geometrie der Lage of 1847, which contains a few gaps). Toepell [1985] notes that

28 The power-of-a-point theorem holds in Euclidean planes (structures with K × K as point set, with K a non-quadratically closed field, in which segment congruence ≡ is defined by (a, b)(c, d) ≡ (a, b)(c, d), if and only if (a − c, b − d) = (a − c, b − d) , with (x, y) =x2 + ky2,wherek is a constant satisfying −k ∈ K 2, with the ‘distance’ between two points defined in terms of · (see [Schroder¨ , 1985]), so the radical-axis concept makes sense for two intersecting circles and the proof proceeds as in the standard Euclidean case. Since every metric-Euclidean plane can be embedded in a Euclidean plane, and the Three-Chord Theorem is a universal statement, it must hold in all metric-Euclidean planes as well. 29 Freudenthal’s criticism of the betweenness relation itself, which he deems old- fashioned, proposing the binary-order relation from algebra, and of the ‘early’ introduction of the order axioms [Freudenthal, 1957, p. 119] is misguided. There are alternatives for the betweenness relation, but no binary one among points, and it is Pasch’s and Hilbert’s ‘early’ introduction of the order axioms that motivated much of the very rich work in ordered geometry, surveyed in [Pambuccian, 2011]. 30 A mathematician whose name Weyl no longer recalled once told Hilbert ‘You have forced us all to consider important those problems you considered important’ [Weyl, 1944, p. 615]. 268 PHILOSOPHIA MATHEMATICA it was Friedrich Schur who first proved a Papp with the help of congru- ence axioms but without the Archimedean axiom [Schur, 1899]and[Weyl, 1944], and that Hilbert’s citation practice leaves much to be desired, not only in Schur’s case.31 Stroppel [2011]andArana and Mancosu [2012] show that the independence of the Desargues theorem from the other inci- dence axioms is implicit in the work of Beltrami going back to 1865, and Downloaded from is explicit in a paper of Peano of 1894. We believe that the significance of the GdG has been largely misunder- stood. Today, most mathematicians would consider it as a work in which

Hilbert presented a modern axiomatization of Euclidean three-dimensional http://philmat.oxfordjournals.org/ geometry, the main aim being to fill the gaps in Euclid. As such, it has been relegated to the museum of classical works, without much connec- tion with current mathematical practice or interests. The widespread belief that Hilbert’s aim was to axiomatize three-dimensional Euclidean geome- try over the real numbers32 has given ‘working mathematicians’ the perfect excuse to ignore the axiomatic point of view,33 as, according to this belief, the axiomatic set-up has little bearing on the results obtained, and work in models is fully justified, as second-order logic rarely offers the option of a syntactic proof (there is no known syntactic use for the final completeness at Arizona State University Libraries on December 3, 2013 axiom V.2). Again, in this widespread view, GdG could have a use as a pedagogical example of the axiomatic method, to be used in the undergraduate math- ematics curriculum or in that of future high-school mathematics teachers. As such, it was found already by G.D. Birkhoff [1932] to be too demand- ing, and was replaced by an axiom system in which the real numbers show up on the first page, in Postulate I ‘of line measure’! This tradition has been followed by most textbooks in use today, if they even pretend to provide an axiom system (many skip that task altogether, perceived as tedious,

31 ‘In his papers one encounters not infrequently utterances of pride in a beautiful or unexpected result and in his legitimate satisfaction he sometimes did not give to his pre- decessors on whose ideas he built all the credit they deserved’ [Weyl, 1944, p. 615]. Like- minded statements can also be found in [Marchisotto and Smith, 2007],andin[Mancosu, 2010, p. 11] we read: ‘In light of the importance of the work of Peano and his school on the foundations of geometry, it is quite surprising that Hilbert did not acknowledge their work in the Foundations of Geometry.’ 32 An entirely erroneous assumption, given that there is no completeness axiom in the Festschrift edition of GdG (it was introduced in the French translation of 1900), as pointed out in [Rowe, 2000,p.69]. 33 As Juliette Kennedy [forthcoming] puts it, ‘That mathematics is practiced in what one might call a formalism free manner has always been the case — and remains the case. Of course, no one would have thought to put it this way prior to the emergence of the foundational formal systems in the late nineteenth and early twentieth centuries.’ CRITICAL STUDIES/BOOK REVIEWS 269 boring, and useless), with two major exceptions: [Greenberg, 2008]and [Hartshorne, 2000]. To think that GdG was about providing an axiom system for school geometry and then to complain that the axiom system is too complex is much like assuming that Don Quixote is children’s literature, and then complaining that it is too long for its intended audience. Downloaded from It has been shown by Robin Hartshorne, who has used [Hartshorne, 2000] as an undergraduate textbook at the University of California at Berkeley, that it is possible to present GdG to a contemporary mathemati- cally mature audience — emphasizing all of the major Hilbertian themes: the continuity with Euclid’s Elements, the avoidance of a symbolic lan- http://philmat.oxfordjournals.org/ guage, the axiomatic introduction of area, the impossibility of construc- tions with certain sets of instruments, the continuity-free axiomatization of in the style of [Hilbert, 1903a], as well as Dehn’s solution to Hilbert’s Third Problem — with complete proofs, and updated to today’s language and algebraic understanding. An alternative approach, stemming from Pieri’s work and that of the Italian school, which expressed axiom systems in symbolic language, is that of Tarski, whose axiom system [Tarski and Givant, 1999]hasthe advantage that it is presented in a very simple language, with only points at Arizona State University Libraries on December 3, 2013 as variables, two predicates as primitive notions, and very few axioms.34 The proofs of all theorems required to reach algebraization are provided in [Schwabhauser,¨ Szmielew, and Tarski, 1983]. There is no doubt that Hilbert’s axiom system, with three types of variables, and a large number of predicates, was not meant to be presented as a formal system (although it has been formalized four times so far, in [Rossler¨ , 1934; Helmer, 1935; Cassina, 1948–1949; Schwabhauser¨ , 1956], and one of the reasons for GdG’s popularity — compared to the muted response Peano’s or Pieri’s work has received — is precisely the natural-language presentation of the material, as pointed out in [Marchisotto and Smith, 2007, pp. 274–277]. In fact, not much has changed since March 31, 1897, when Felix Klein wrote to Mario Pieri that he would accept a survey of his results for publication in the Mathematische Annalen only if it was not written in formal language, since papers ‘written in this symbolic language, at least in , find practically no readers, but rather stumble upon rejection from the start’.35

34 However, in matters of simplicity, many mathematicians consider, in the spirit of the Rumanian saying that ‘The shortest road is the known road’, that the simplest axiomatiza- tion is the known one, i.e., one that most resembles Hilbert’s. The reviewer has provided for the past 30 years various criteria for simplicity and has provided absolutely simplest axiom systems according to multiple criteria, only to hear that the proposed axiom systems are not at all simple... 35 ‘Meine allgemeine Erfahrung besteht namlich¨ darin, dass Arbeiten, welche in dieser Symbolik geschrieben sind, jedenfalls in Deutschland, so gut wie keine Leser finden, vielmehr von vornherein auf Ablehnung stossen.’ [Luciano and Roero, 2012, p. 188] 270 PHILOSOPHIA MATHEMATICA

As shown in (ii), Hilbert managed to turn a weakness — the many predicates the language has — into a strength, by asking the questions referred to earlier, many of which asked for the strength of a certain fragment of geometry, expressed in a language obtained by dropping some of the primitive notions. Even the use of the notion of angle- congruence, which had already been shown to be dispensable in [Mollerup, Downloaded from 1904], turned out to create both apparently intractable problems (such as the question whether the ‘Side-Angle-Side’ congruence axiom can be replaced by ‘Side-Side-Side’ or by ‘Side-Angle-Angle’ in the frame- work of Hilbert’s absolute geometry (assuming the completeness axiom,

‘Side-Angle-Angle’ was shown in [Donnelly, 2010]) that do not exist in http://philmat.oxfordjournals.org/ Tarski’s axiom system, as well as axiomatizations using angle-congruence but not segment-congruence [Sch¨utte, 1955b; Quaisser, 1973; Schaeffer, 1979; Pambuccian, 1998b]. ThetruevalueofGdG can be found in the unprecedented depth with which it treats its subject, in the stupefying magic of its models, even those that appeared at the time to be clumsy, like the model of inde- pendence for Desargues axiom in the 1898–99 lectures, which popped up in the classification of R2-planes with a 3-dimensional group almost eighty years later in [Betten and Ostmann, 1978] (see [Stroppel, 1998]), or at Arizona State University Libraries on December 3, 2013 the model of independence of a Des that appeared in the Festschrift edi- tion, that is still of interest today [Anisov, 1992; Schneider and Stroppel, 2007]. The lectures only deepen that awe at Hilbert’s ability to create mod- els, and reading [Hilbert, 1903b] one has no doubt that, despite having had illustrious predecessors, by becoming the architect of barely possible worlds, Hilbert turned the foundations of geometry into something entirely different. If we ask what the legacy of GdG and, more generally, of the axiomatic foundation of geometry is, we find that there are two contrasting legacies. The first is in the written record, and it is astonishing, with more than 1300 papers and monographs devoted to the axiomatic founda- tion of geometry. These solve a wide range of problems and pro- vide elementary axiomatizations for several elementary fragments of the geometries invented in the nineteenth century (the most impor- tant monographs being [Pickert, 1975; Bachmann, 1973; Benz, 1973; Schwabhauser,¨ Szmielew, and Tarski, 1983]. We also find, quite surpris- ingly, that the axiomatic foundation of geometry has remained for the most part rooted geographically in the two countries in which research in the axiomatic foundations of geometry was active in the late nineteenth cen- tury: Germany and Italy. The major exception is Poland, where research in the axiomatic foundations of geometry has been intense, and where there was no nineteenth-century contribution, the interest in this endeavor orig- inating with . In the United States there was a short-lived interest in axiomatics, around, Eliakim H. Moore Oswald, Veblen and CRITICAL STUDIES/BOOK REVIEWS 271

Edward V. Huntington, that ended in 1930 (see [Scanlan, 1991]). Later Tarski showed considerable interest, but had very few followers in the United States. Karl Menger and his students (from Notre Dame and the Illinois Institute of Technology) provided surprisingly simple incidence- based axiom systems for plane hyperbolic geometry and for some of its fragments. Independent lines of research were pursued in the work of Downloaded from Michael Kallaher, James T. Smith, Marvin Greenberg, Robin Hartshorne, Michael Beeson, and Robert Knight. Mathematicians in France, Russia (or the Soviet Union), Japan, and the United Kingdom showed by and large no interest, and had a low regard for the field if they were aware of its existence. http://philmat.oxfordjournals.org/ In the collective memory of ‘working mathematicians’, however, there is no awareness of the axiomatic foundations of geometry as a field of research with its own challenging problems, entirely unrelated to textbook presentations of axiom systems, which was opened up by the work of Pasch, Peano, Pieri, Schur, and decisively influenced by Hilbert and the GdG.36 Unless they have worked in the axiomatic foundations of geome- try, mathematicians tend to be oblivious to the 130 years of research in this area that have passed since the publication of Moritz Pasch’s Vorlesungen uber¨ neuere Geometrie, and believe that GdG is the crowning achievement at Arizona State University Libraries on December 3, 2013 of this axiomatizing endeavor.

References Abellanas, Pedro [1946]: ‘Estructura analitica del segmento abierto definido por los postulados de incidencia y orden de Hilbert’, Revista Matematica´ Hispano-Americana (4) 6, 101–126. Andreka,´ Hajnal, Judit X. Madarasz´ , and Istvan´ Nemeti´ [2006]: ‘Logical axiomatizations of space-time. Samples from the literature’, in A. Prekopa´ and E. Molnar,´ eds, Non-Euclidean geometries, pp. 155–185. New York: Springer. Andreka,´ Hajnal, Judit X. Madarasz´ , Istvan´ Nemeti´ , and Gergely Szekely´ [2008]: ‘Axiomatizing relativistic dynamics without conservation postulates’, Studia Logica 89, 163–186. Anisov, Sergei S. [1992]: ‘The collineation group of Hilbert’s example of a pro- jective plane’, Uspekhi Matematicheskikh Nauk 47, no. 3 (285), 147–148; translation in Russian Mathematical Surveys 47, 163–164. Arana, Andrew [2008]: ‘Logical and semantic purity’ in G. Preyer and G. Peter, eds, Philosophy of mathematics. Set theory, measuring theories, and nomi- nalism, pp. 40–52. : Ontos Verlag. Arana, Andrew, and Michael Detlefsen [2011]: ‘Purity of methods’, Philosophers’ Imprint 11, no. 2, 1–20.

36 There may be some awareness of the more active areas that have foundational charac- ter, such as finite geometries or buildings, but little of it is of a first-order axiomatic nature ([Shult, 2011] comes closest to an axiomatization in Hilbert’s sense). 272 PHILOSOPHIA MATHEMATICA

Arana, Andrew, and Paolo Mancosu [2012]: ‘On the relationship between plane and solid geometry’, Review of Symbolic Logic 5, 294–353. Artin, Emil, and Otto Schreier [1926]: ‘Algebraische Konstruktion reeller Korper’,¨ Abhandlungen aus dem mathematischen Seminar der Universitat¨ Hamburg 5, 85–99. Bachmann, Friedrich [1951]: ‘Zur Begr¨undung der Geometrie aus dem Downloaded from Spiegelungsbegriff’, Math. Ann. 123, 341–344. ——— [1967]: ‘Der Hohensatz¨ in der Geometrie involutorischer Gruppenele- mente’, Canadian Journal of Mathematics 19, 895–903. ——— [1973]: Aufbau der Geometrie aus dem Spiegelungsbegriff. Zweite erganzte¨ Auflage. Berlin: Springer-Verlag.

Baldwin, John T. [2013]: ‘Formalization, primitive concepts, and purity’, http://philmat.oxfordjournals.org/ Review of Symbolic Logic 6, 87–128. Barbilian, Dan [1936]: ‘Exkurs ¨uberdie Dreiecke’, Bulletin Mathematique´ de la Societ´ e´ des Sciences Mathematiques´ de Roumanie 38, 3–62. Benz, Walter [1973]: Vorlesungen uber¨ Geometrie der Algebren. Geometrien von Mobius,¨ Laguerre-Lie, Minkowski in einheitlicher und grundlagenge- ometrischer Behandlung. Berlin: Springer-Verlag. Betten, Dieter, and Axel Ostmann [1978]: ‘Wirkungen und Geometrien der Gruppe L2 × R’, Geometriae Dedicata 7, 141–162. Birkhoff, George David [1932]: ‘A set of postulates for plane geometry based on scale and protractor’, Annals of Mathematics 33, 329–345. at Arizona State University Libraries on December 3, 2013 Bruck, Richard H., and Erwin Kleinfeld [1951]: ‘The structure of alterna- tive division rings’, Proceedings of the American Mathematical Society 2, 878–890. Cassina, Ugo [1948–1949]: ‘Ancora sui fondamenti della geometria secondo Hilbert. I, II, III’, Istituto Lombardo di Scienze e Lettere. Rendiconti, Classe di Scienze Matematiche e Naturali (3) 12 (81), 71–94; 13 (82), 67–84, 85–94. Corry, Leo [2000]: ‘The empiricist roots of Hilbert’s axiomatic approach’, in V.F. Hendricks, S.A. Pedersen and K.F. Jørgensen, eds, Proof theory (Roskilde, October 31–November 1, 1997), pp. 35–54. Dordrecht: Kluwer. ——— [2002]: ‘David Hilbert y su filosof´ıa empiricista de la geometr´ıa’, Bolet´ın de la Asociacion´ Matematica´ Venezolana 9, 27–43. ——— [2006]: ‘Axiomatics, empiricism, and Anschauung in Hilbert’s concep- tion of geometry: Between arithmetic and general relativity’, in J. Ferreiros´ and J.J. Gray, eds, The Architecture of Modern Mathematics, pp. 133–156. Oxford: Oxford University Press. Cronheim, Arno [1953]: ‘A proof of Hessenberg’s theorem’, Proceedings of the American Mathematical Society 4, 219–221. Dehn, Max [1900]: ‘Die Legendre’schen Satze¨ uber ¨ die Winkelsumme im Dreieck’, Mathematische Annalen 53, 404–439. Ferreiros,´ Jose´ [2009]: ‘Hilbert, logicism, and mathematical existence’, Syn- these 170, 33–70. Donnelly, John [2010]: ‘The equivalence of side-angle-side and side-angle- angle in the absolute plane’, Journal of Geometry 97, 69–82. Freudenthal, Hans [1957]: ‘Zur Geschichte der Grundlagen der Geome- trie. Zugleich eine Besprechung der 8. Aufl. von Hilberts “Grundlagen der Geometrie” ’, Nieuw Archief voor Wiskunde (3) 5, 105–142. CRITICAL STUDIES/BOOK REVIEWS 273

Fritsch, Rudolf [1974]: ‘Synthetische Einbettung Desarguesscher Ebenen in Raume’,¨ Mathematisch-Physikalische Semesterberichte 21, 237–249. ——— [1995]: ‘Ein axiomatischer Zugang zu einigen Winkelsatzen¨ der ebenen Geometrie’, Schriften der Sudetendeutschen Akademie der Wissenschaften und Kunste.¨ Forschungsbeitrage¨ der Naturwissenschaftlichen Klasse 16, 41– 57. Downloaded from Gandon, Sebastien´ [2005]: ‘Pasch entre Klein et Peano: empirisme et idealit´ e´ en geom´ etrie’,´ Dialogue 44, 653–692. Greenberg, Marvin Jay [2008]: Euclidean and Non-Euclidean Geometries.4th edition. San Francisco: W.H. Freeman. ——— [2010]: ‘Old and new results in the foundations of elementary plane

Euclidean and non-Euclidean geometries’, American Mathematical Monthly http://philmat.oxfordjournals.org/ 117, 198–219. Guicciardini, Niccolo` [2009]: Isaac Newton on Mathematical Certainty and Method. Cambridge, Mass.: MIT Press. Gupta, Haragauri Narayan [1965]: Contributions to the Axiomatic Foun- dations of Euclidean Geometry. Ph.D. Thesis, University of California, Berkeley. Hartshorne, Robin [2000]: Geometry: Euclid and Beyond. New York: Springer- Verlag. ——— [2003]: ‘Non-Euclidean III.36’, American Mathematical Monthly 110, 495–502. at Arizona State University Libraries on December 3, 2013 Helmer-Hirschberg, Olaf [1935]: ‘Axiomatischer Aufbau der Geometrie in formalisierter Darstellung’, Schriften des mathematische Seminars und des Instituts fur¨ angewandte Mathematik der Universitat¨ Berlin 2, 175–201. Herzer, Armin [1975]: ‘Neue Konstruktion einer Erweiterung von projektiven Geometrien’, Geometriae Dedicata 4, 199–213. Hessenberg, Gerhard [1905a]: ‘Beweis des Desargues’ schen Satzes aus dem Pascal’, schen’, Mathematische Annalen 61, 161–172. ——— [1905b]: ‘Neue Begr¨undungder Spharik’,¨ Sitzungsberichte der Berliner Mathematischen Gesellschaft 4, 69–77. Hilbert, David [1903a]: ‘Neue Begr¨undung der Bolyai-Lobatschefsky schen Geometrie’, Mathematische Annalen 57, 137–150. ——— [1903b]: ‘Uber¨ den Satz von der Gleichheit der Basiswinkel im gleich- schenkligen Dreieck’, Proceedings of the London Mathematical Society 35, 50–68. ——— [1971]: Les fondements de la geom´ etrie.´ Edition´ critique avec introduc- tion et complements´ prepar´ ee´ par Paul Rossier. Paris: Dunod. Hjelmslev, Johannes [1907]: ‘Neue Begr¨undungder ebenen Geometrie’, Math- ematische Annalen 64, 449–474. ——— [1929]: ‘Einleitung in die allgemeine Kongruenzlehre. I, II.’, Det Kon- gelige Danske Videnskabernes Selskab Matematisk-Fysiske Meddelelser 8, No. 11, 1–36; 10, No. 1, 1–28. Hociota,˘ Ioana, and Victor Pambuccian [2011]: ‘Acute triangulation of a tri- angle in a general setting revisited’, Journal of Geometry 102, 81–84. Joussen, Jakob [1966]: ‘Die Anordnungsfahigkeit¨ der freien Ebenen’, Abhand- lungen aus dem Mathematischen Seminar der Universitat¨ Hamburg 29, 137–184. 274 PHILOSOPHIA MATHEMATICA

Kalhoff, Franz [1988]: ‘Eine Kennzeichnung anordnungsfahiger¨ Ternar-¨ korper’,¨ Journal of Geometry 31, 100–113. Karzel, Helmut, and Hans-Joachim Kroll [1988]: Geschichte der Geometrie seit Hilbert. Darmstadt: Wissenschaftliche Buchgesellschaft. Kennedy, Juliette [forthcoming]: ‘On formalism freeness’, Bulletin of Sym- bolic Logic. Downloaded from Klev, Ansten [2011]: ‘Dedekind and Hilbert on the foundations of the deductive sciences’, Review of Symbolic Logic 4, 645–681. Kusak, Eugeniusz [1987]: ‘Desarguesian Euclidean planes and their axiom sys- tem’, Bulletin of the Polish Academy of Sciences: Mathematics 35, 87–92. Levi, Friedrich Wilhelm [1939]: ‘On a fundamental theorem of geometry’,

Journal of the Indian Mathematical Society (New Series) 3, 182–192. http://philmat.oxfordjournals.org/ Lingenberg, Rolf [1959, 1960/1961, 1965]: ‘Uber¨ Gruppen mit einem invari- anten System involutorischer Erzeugender, in dem der allgemeine Satz von den drei Speigelungen gilt. I, II, III, IV’, Mathematische Annalen 137, 26–41, 83–106; 142, 184–224; 158, 297–325. ——— [1979]: Metric planes and metric vector spaces. New York: Wiley- Interscience. Luciano, Erika, and Clara Silvia Roero [2012]: ‘From Turin to Gottingen:¨ dialogues and correspondence (1879–1923)’, Bollettino di Storia delle Scienze Matematiche 32, 9–232. Madarasz,´ Judit X., Istvan´ Nemeti´ , and Gergely Szekely´ [2006]: ‘Twin at Arizona State University Libraries on December 3, 2013 paradox and the logical foundation of relativity theory’, Foundations of Physics 36, 681–714. Makarios, Timothy James McKenzie [2013]: ‘A further simplification of Tarski’s axioms of geometry’. Preprint. Mancosu, Paolo [2010]: The Adventure of Reason. Interplay Between Philos- ophy of Mathematics and , 1900–1940. Oxford: Oxford University Press. Marchisotto, Elena Anne, and James T. Smith [2007]: The Legacy of Mario Pieri in Geometry and Arithmetic. Boston: Birkhauser.¨ Mollerup, Johannes [1904]: ‘Die Beweise der ebenen Geometrie ohne Benutzung der Gleichheit und Ungleichheit der Winkel’, Mathematische Annalen 58, 479–496. Moufang, Ruth, [1931]: ‘Die Einf¨uhrungder idealen Elemente in die ebene Geometrie mit Hilfe des Satzes vom vollstandigen¨ Vierseit’, Mathematische Annalen 105, 759–778. Naumann, Heribert [1956]: ‘Eine affine Rechtwinkelgeometrie’, Mathema- tische Annalen 131, 17–27. Naumann, Heribert, and Kurt Reidemeister [1957]: ‘Uber¨ Schließungssatze¨ der Rechtwinkelgeometrie’, Abhandlungen aus dem mathematischen Semi- nar der Universitat¨ Hamburg 21, 1–12. Newton, Isaac [1673/83]: The Mathematical Papers of Isaac Newton.Vol.5. D.T. Whiteside, ed. Cambridge: Cambridge University Press, 1972. Owens, Frederick William [1910]: ‘The introduction of ideal elements and a new definition of projective n-space’, Transactions of the American Mathe- matical Society 11, 141–171. CRITICAL STUDIES/BOOK REVIEWS 275

Pambuccian, Victor [1998]: ‘Zur Existenz gleichseitiger Dreiecke in H- Ebenen’, Journal of Geometry 63, 147–153. ——— [1998]: ‘Zur konstruktiven Geometrie euklidischer Ebenen’, Abhandlun- gen aus dem mathematischen Seminar der Universitat¨ Hamburg 68, 7–16. ——— [2001]: ‘A methodologically pure proof of a convex geometry problem’, Beitrage¨ zur Algebra und Geometrie 42, 401–406. Downloaded from ——— [2003]: ‘On the planarity of the equilateral, isogonal pentagon’, Mathe- matica Pannonica 14, 101–112. ——— [2005a]: ‘Euclidean geometry problems rephrased in terms of midpoints and point-reflections’, Elemente der Mathematik 60, 19–24. ——— [2005b]: ‘Groups and plane geometry’, Studia Logica 81, 387–398.

——— [2006]: ‘Positive definitions of segment congruence in terms of segment http://philmat.oxfordjournals.org/ inequality’, Aequationes Mathematicae 72, 243–253. ——— [2007a]: ‘Orthogonality as single primitive notion for metric planes’, with an appendix by Horst and Rolf Struve, Beitrage¨ zur Algebra und Geometrie 48, 399–409. ——— [2007b]: ‘Alexandrov-Zeeman type theorems expressed in terms of defin- ability’, Aequationes Mathematicae 74, 249–261. ——— [2009a]: ‘On a paper of Dan Barbilian’, Note di Matematica 29, 29–31. ——— [2009b]: ‘A reverse analysis of the Sylvester-Gallai theorem’, Notre Dame Journal of Formal Logic 50, 245–260. ——— [2010]: ‘Acute triangulation of a triangle in a general setting’, Canadian at Arizona State University Libraries on December 3, 2013 Mathematical Bulletin 53, 534–541. ——— [2011]: ‘The axiomatics of ordered geometry. I. Ordered incidence spaces’, Expositiones Mathematicae 29, 24–66. ——— [2012]: ‘An axiomatic look at a windmill’, http://arxiv.org/pdf/1209. 5979.pdf. Pambuccian, Victor, and Rolf Struve [2009]: ‘On M.T. Calapso’s characteri- zation of the metric of an absolute plane’, Journal of Geometry 92, 105–116. Pambuccian, Victor, Horst Struve, and Rolf Struve [forthcoming]: ‘The Steiner-Lehmus theorem and “triangles with congruent medians are isosce- les” hold in weak geometries’, Journal of Geometry. Pasch, Moritz [1882]: Vorlesungen uber¨ neuere Geometrie. Leipzig: B.G. Teubner. Pejas, Wolfgang [1961]: ‘Die Modelle des Hilbertschen Axiomensystems der absoluten Geometrie’, Mathematische Annalen 143, 212–235. Pickert, Gunter¨ [1951]: Einfuhrung¨ in die hohere¨ Algebra.Gottingen:¨ Vanden- hoeck & Ruprecht. ——— [1975]: Projektive Ebenen. Berlin: Springer-Verlag. Quaisser, Erhard [1973]: ‘Winkelmetrik in affin-metrischen Ebenen’, Zeitschrift fur¨ mathematische Logik und Grundlagen der Mathematik 19, 17–32. ——— [1975]: ‘Zu einer Orthogonalitatsrelation¨ in Desarguesschen Ebenen (der Charakteristik = 2)’, Beitrage¨ zur Algebra und Geometrie 4, 71–84. Rossler,¨ Karel [1934]: ‘Geom´ etrie´ abstraite mecanis´ ee’,´ Publications de la Faculte´ des Sciences de l’Universite´ Charles (Praha) 134. 1–29. Rowe, David [2000]: ‘The calm before the storm: Hilbert’s early views on foun- dations’, in V.F. Hendricks, S.A. Pedersen and K.F. Jørgensen, eds, Proof 276 PHILOSOPHIA MATHEMATICA

theory (Roskilde, October 31–November 1, 1997), pp. 55–83. Dordrecht: Kluwer. Saad, Gerhard [1988]: Beitrage¨ zur Theorie metrischer Keime. Dissertation, Fachbereich Mathematik, Universitat¨ Hamburg. Scanlan, Michael [1991]: ‘Who were the American postulate theorists?’, Jour- nal of Symbolic Logic 56, 981–1002. Downloaded from Schaeffer, Helmut [1979]: ‘Eine Winkelmetrik zur Begr¨undungeuklidisch- metrischer Ebenen’, Journal of Geometry 12, 152–167. Schlimm, Dirk [2010]: ‘Pasch’s philosophy of mathematics’, Review of Symbolic Logic 3, 93–118. Schmidt, Arnold [1943]: ‘Die Dualitat¨ von Inzidenz und Senkrechtstehen in der

absoluten Geometrie’, Mathematische Annalen 118, 609–625. http://philmat.oxfordjournals.org/ Schneider, Thomas, and Markus Stroppel: ‘Automorphisms of Hilbert’s non- Desarguesian affine plane and its projective closure’, Advances in Geometry 7, 541–552. Schor, Dimitry [1904]: ‘Neuer Beweis eines Satzes aus den “Grundlagen der Geometrie” von Hilbert’, Mathematische Annalen 58, 427–433. Schroder,¨ Eberhard M. [1984]: ‘Aufbau metrischer Geometrie aus der Hexa- grammbedingung’, Atti del Seminario Matematico e Fisico dell’Universitadi` Modena 33, 183–217. ——— [1985]: Geometrie euklidischer Ebenen. Mathematische Grundlegung der Schulgeometrie. Paderborn: Ferdinand Schoningh.¨ at Arizona State University Libraries on December 3, 2013 Schur, Friedrich [1899]: ‘Uber¨ den Fundamentalsatz der projectiven Geome- trie’, Mathematische Annalen 51, 401–409. Schutte,¨ Kurt [1955a]: ‘Ein Schließungssatz f¨ur Inzidenz und Orthogonalitat’,¨ Mathematische Annalen 129, 424–430. ——— [1955b]: ‘Die Winkelmetrik in der affin-orthogonalen Ebene’, Mathema- tische Annalen 130, 183–195. Schwabhauser,¨ Wolfram [1956]: ‘Uber¨ die Vollstandigkeit¨ der elementaren euklidischen Geometrie’, Zeitschrift fur¨ mathematische Logik und Grund- lagen der Mathematik 2, 137–165. Schwabhauser,¨ Wolfram, Wanda Szmielew, and Alfred Tarski [1983]: Metamathematische Methoden in der Geometrie. Berlin: Springer-Verlag. Re-issue: Bronx, N.Y.: Ishi Press International, 2011. Shult, Ernest E. [2011]: Points and Lines. Characterizing the Classical Geome- tries. New York: Springer. Skornyakov, Lev A. [1949]: ‘Natural domains of Veblen-Wedderburn projective planes’, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 13, 447– 472. English translation in: American Mathematical Society Translations 58 (1951), 1–37. Sperner, Emanuel [1938]: ‘Zur Begr¨undung der Geometrie im begrenzten Ebenenst¨uck’, Schriften der Konigsberger¨ gelehrten Gesellschaft: Naturwis- senschaftliche Klasse 14, No. 6, 121–143. ——— [1954]: ‘Ein gruppentheoretischer Beweis des Satzes von Desargues in der absoluten Axiomatik’, Archiv der Mathematik (Basel) 5, 458–468. Sorensen,¨ Kay [1984]: ‘Ebenen mit Kongruenz’, Journal of Geometry 22, 15–30. BOOKS OF ESSAYS 277

Stroppel, Markus [1998]: ‘Bemerkungen zur ersten nicht Desarguesschen ebe- nen Geometrie bei Hilbert’, Journal of Geometry 63, 183–195. ——— [2011]: ‘Early explicit examples of non-Desarguesian plane geometries’, Journal of Geometry 102, 179–188. Struve, Horst, and Rolf Struve [1985]: ‘Eine synthetische Charakterisierung der Cayley-Kleinschen Geometrien’, Zeitschrift fur¨ mathematische Logik Downloaded from und Grundlagen der Mathematik 31, 569–573. ——— [2004]: ‘Projective spaces with Cayley-Klein metrics’, Journal of Geom- etry 81, 155–167. ——— [2010]: ‘Non-Euclidean geometries: The Cayley-Klein approach’, Jour- nal of Geometry 98, 151–170.

Szekely,´ Gergely [2010]: ‘A geometrical characterization of the twin paradox http://philmat.oxfordjournals.org/ and its variants’, Studia Logica 95, 161–182. Szele, Tibor´ [1952]: ‘On ordered skew fields’, Proceedings of the American Mathematical Society 3, 410–413. Szmielew, Wanda [1983]: From Affine to Euclidean Geometry. Dordrecht: D. Reidel; : PWN-Polish Scientific Publishers. Tarski, Alfred, and Steven Givant [1999]: ‘Tarski’s system of geometry’, Bul- letin of Symbolic Logic 5, 175–214. Toepell, Michael-Markus [1985]: ‘Zur Schl¨usselrolle Friedrich Schurs bei der Entstehung von David Hilberts “Grundlagen der Geometrie” ’, in M. Folkerts and U. Lindgren, eds, Mathemata. Festschrift fur¨ Helmuth Gericke, pp. 637– at Arizona State University Libraries on December 3, 2013 649. Wiesbaden: Franz Steiner. ——— [1986]: Uber¨ die Entstehung von David Hilberts “Grundlagen der Geometrie”.Gottingen:¨ Vandenhoeck & Ruprecht. Weyl, Hermann [1944]: ‘David Hilbert and his mathematical work’, Bulletin of the American Mathematical Society 50, 612–654. Wiener, Hermann [1893]: Sechs Abhandlungen uber¨ das Rechnen mit Spiegelungen, nebst Anwendungen auf die Geometrie der Bewegungen und auf die projective Geometrie. Leipzig: Breitkopf & Hartel.¨ doi:10.1093/phimat/nkt014 Advance Access publication May 10, 2013

Books of Essays Mircea Pitici, ed. The Best Writing on Mathematics 2011. Princeton: Prince- ton University Press, 2012. ISBN: 978-0-691-15315-5 (pbk); 978-1-400-83954-4 (e-book). Pp. xxx + 384.

AUTHORS AND TITLES

Freeman Dyson, Foreword: Recreational mathematics, pp. xi–xvi. Mircea Pitici, Introduction, pp. xvii–xxx. Underwood Dudley, What Is mathematics for?, pp. 1–12. Dana Mackenzie, A tisket, a tasket, an Apollonian gasket, pp. 13–26. Rik van Grol, The quest for God’s number, pp. 27–34.